I got a question why when f has constrained extreme values, (grad)f and (grad)g are parallel.

The answer for this question is this(the proof is on the page 226 of the textbook).

For a curve c(t) contained in the level surface g(x)=c and having c(0)=x0,  
                           dg(c(t))/dt = (grad)g(x0) ·  c'(0) = 0

Moreover, if f has its extreme value at this point x0, then
                           df(c(t))/dt = (grad)f(x0) ·  c'(0) =0
                          
  =>as a function of t, if f has its extreme value at t=0, then its derivative is obviously zero at t=0.

So, (grad)f and (grad)g are perpendicular to the tangent plane at the point x0.

So they are parallel to each other=>(grad)f= λ(grad)g

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